I understand that a clock on the surface of the Earth is observed to run more slowly than a clock in orbit or a clock "at infinity". What about a clock situated at the center of the earth. Would it be slowed because of the concentration of mass? Or would it run normal speed because there is no...
No. An inconsistent system can prove any statement. If (system_2) is inconsistent it can prove (system_1) is consistent. For what it's worth, if (system_2) is inconsistent, it can also prove (system_1) is inconsistent.
Are you sure about that? We have zero and one, the additive and multiplicative identities. Using 1 and addition, we can define a successor function, f(x) = x+1. We can define N as the smallest set that contains 0 and for every n in N, we also find n+1 in N.
I'm pretty sure that we cannot...
Godel showed how to translate statements about a formal system into statements about integers. The formal system can be the integers or the reals. The translation does not depend upon the idea of a successor in the system being modeled. Godel's Incompleteness Theorem applies to any formal system...
The real numbers form a complete ordered field. This use of complete is not the complete used by Godel. The axioms of the real numbers are not complete in the sense Godel uses the word complete.
The reals and the integers are not constructed. The real numbers is a set which has certain properties. The set of integers is a set with certain other properties. You can construct a model for the integers using set theory. You can construct a model for the real numbers from the integers. The...
The co-domain for the function in the first question cannot be a proper subset of [0,16] or you would not have a function. Where you said "is a subset of" should be replaced with "equals" for both questions.
Well, yes and no. I hope you're not one of those who think scientific theory is "just a guess". A theory explains obervations and makes testable predictions. Relativity explains observations and makes predictions. Every test of the theory has supported relativity. Unless someone can come up with...
Eh. . . The acceleration is constant. your post should read: "If you find the first derivative of distance as a function of time this would give you a formula for velocity, in this case it would be d'(t) = 36t."
Oops. Two coordinates are needed to specify an axis. You'd need a third to specify rotation about that axis. Sorry. But I maintain rotational coordinates are not physical dimensions.
You only need 2 coorinates to specify rotation of a 3d object. Consider an arrow at the center of a shpere pointing north. You only need latitude and longitude to specify any rotated position. But, while rotation could be considered dimensions in a phase space, they are not physical dimensions...